A construction of the quantum Steenrod squares and their algebraic relations (Q2004516)

In algebraic topology, the Steenrod squares of cohomology classes are algebraic operations uniquely defined by a series of axioms. It is noteworthy that the uniqueness does not include a canonical construction of these operations. The main theme of this long paper stems from symplectic geometry. From an algebraic topology point of view, a natural problem in symplectic geometry is how to construct a quantum version of Steenrod squares for quantum cohomology classes of a symplectic manifold. Differing from the classical case, the quantum Steenrod square is not necessarily axiomatically defined, but only a construction which was first suggested by Fukaya based on his Morse homotopy theory. In the paper under review, the author gives a new construction of quantum Steenrod squares which differs from the one constructed by Fukaya. The primary objects of the study of this paper are the following: (1) The author shows that the Adem and Cartan relations fail for quantum Steenrod squares and give their quantum deformations called the quantum Cartan relation (Theorem 1.2) and quantum Adem relation (Corollary 1.8), which solves an open problem proposed by \textit{K. Fukaya} [AMS/IP Stud. Adv. Math. 2, 409--440 (1997; Zbl 0891.57035)]. (2) For certain closed monotone symplectic manifolds, the author shows the explicit computations and the solution to the quantum deformation of Steenrod squares, such as toric varieties. As applications, the author considered the quantum Steenrod square for blow-ups of varieties (Section 8).